In this paper, research on transverse vibrations of axially moving strings and their control is thoroughly reviewed. In the last few decades, there have been extensive studies on analysis and control of transverse vibrations of axially moving strings because of the wide applications of many engineering devices that axially moving strings represent. In the investigations adopting linear models of moving strings, the paper summarizes recent studies on modal analysis, complicatedly constrained strings, coupled vibrations, and parametric vibration, as well as some early results. In the investigations adopting nonlinear models of moving strings, the paper presents the governing equations with large amplitude, and reviews progress on discretized or direct approximate analytical analyses and numerical approaches based on the Galerkin method or the finite difference method. Furthermore, investigations are reported on modeling of damping mechanisms as viscoelastic materials, coupled vibration of power transmission systems, and bifurcation and chaos. The state of the art of active control of moving strings is surveyed on controllability and observability, the Laplace transform domain analysis and the energy analysis, nonlinear vibration control and adaptive vibration control. Finally, future research directions are suggested such as nonlinear vibration of moving strings under complex constraints and couplings, energetics of nonlinear and time-varying strings, bifurcation and chaos in transverse motion of moving strings, control of hybrid systems containing moving strings, robust and adaptive controls of nonlinear moving strings, and experimental investigations. In this review article there are 242 references cited. @DOI: 10.1115/1.1849169#

Analysis and Control of Transverse Vibrations of Axially Moving Strings

The present paper analyzes the dynamic behavior of a simply supported beam subjected to an axial transport of mass. The Galerkin method is used to discretize the problem; a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities is obtained. The system is studied in the sub and super-critical speed ranges with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning the convergence of the series expansion, linear subcritical behavior, bifurcation analysis and stability, and direct simulation of global postcritical dynamics. A homoclinic orbit is found in a high dimensional phase space and its stability and collapse are studied.

Nonlinear dynamics and bifurcations of an axially moving beam

This paper reviews fundamental investigations of band saw vibration and stability. The literature is spread out in journals and reports in a variety of languages, and consklerable duplication of effort exists. It is this diffusion of references which makes a review necessary. The authors have reviewed all pertinent papers known to them, and apologize to both authors and readers for significant work that is missing. It should be noted that a critical review such as this is necessarily tinged by the authors' interpretation of what is fundamental knowledge in a particular field. The review has been limited to topics of direct, ftmdamental importance to band saw vibration and stability. Topics in which vibration is not the main consideration, such as tooth design, power requirements, fatigue cracking, etc., are not explicitly discussed here. Band saw vibration belongs to a class of vibration problems, called axially moving material vibrations, first investigated by Rudolf Skutch [1897]. This class contains research on magnetic and paper tape vibrations [Miranker 1960], moving belt dynamics [Chubachi 1957, 1958 ; Doyle, Hornung 1969; Rhodes 1970; Lai, Chen 1971], chain drive vibrations [Mahalhlgam 1957], vibration of pipes transporting fhlids [Housner 1952; Niordson 1953; Naguleswaran, Williams 1968b: Thurman, Mote 1969b; Liu, Mote 1974: and many others], moving strings and fibers [Sack 1954; Archibald, Emslie 1958: Swope, Ames 1963: and others], cables and catenaries [Simpson 1972], and of course band saw vibrations. The problems listed above are analogous as a first approximation, because the linearized equation of motion and boundary con-

Principal developments in band saw vibration and stability research

The forced non-linear vibrations of an axially moving beam fitted with an intra-span spring-support are investigated numerically in this paper. The equation of motion is obtained via Hamilton’s principle and constitutive relations. This equation is then discretized via the Galerkin method using the eigenfunctions of a hinged-hinged beam as appropriate basis functions. The resultant non-linear ordinary differential equations are then solved via either the pseudo-arclength continuation technique or direct time integration. The sub-critical response is examined when the excitation frequency is set near the first natural frequency for both the systems with and without internal resonances. Bifurcation diagrams of Poincare maps obtained from direct time integration are presented as either the forcing amplitude or the axial speed is varied; as we shall see, a sequence of higher-order bifurcations ensues, involving periodic, quasi-periodic, periodic-doubling, and chaotic motions.

Stability and bifurcations of an axially moving beam with an intermediate spring support

The thermo-mechanical nonlinear dynamics of a buckled axially moving beam is numerically investigated, with special consideration to the case with a three-to-one internal resonance between the first two modes. The equation of motion of the system traveling at a constant axial speed is obtained using Hamilton’s principle. A closed form solution is developed for the post-buckling configuration for the system with an axial speed beyond the first instability. The equation of motion over the buckled state is obtained for the forced system. The equation is reduced into a set of nonlinear ordinary differential equations via the Galerkin method. This set is solved using the pseudo-arclength continuation technique to examine the frequency response curves and direct-time integration to construct bifurcation diagrams of Poincare maps. The vibration characteristics of the system at points of interest in the parameter space are presented in the form of time histories, phase-plane portraits, and Poincare sections.

Thermo-mechanical nonlinear dynamics of a buckled axially moving beam

The three-dimensional non-linear motion of a traveling string is treated in general. The occurrence of singularities and the breakdown of solution are discussed. Elliptic ballooning is investigated, and the condition for the existence of solution for an equilibrium state is analyzed.

Three-dimensional non-linear vibration of a traveling string

Motion of elliptic ballooning for a traveling string at equilibrium is treated in detail. Some invariants of motion are derived. General solutions for axial velocity of the string and the velocity of wave propagation, based on a transformation in terms of the particle function, are presented. A set of particular solutions is then sought to investigate physical aspects of the problem. Properties of solutions for the transverse displacement in the characteristic plane are analyzed.

L.Y. Shih

Papers

Three-dimensional non-linear vibration of a traveling string

Motion of elliptic ballooning for a traveling string